reserve L for non empty addLoopStr;
reserve a,b,c,x for Element of L;
reserve L for non empty multLoopStr;
reserve a,b,c,x,y,z for Element of L;
reserve G for multGroup;
reserve a,b,c,x for Element of G;
reserve G for associative almost_invertible almost_cancelable well-unital non
  empty multLoopStr_0;
reserve a,x for Element of G;

theorem
  a<>0.G implies a"*a=1.G & a*(a") = 1.G
proof
  assume
A1: a<>0.G;
  hence
A2: a"*a = 1.G by Def10;
  consider x such that
A3: a*x = 1.G by A1,Def8;
  a"*a*x = a" * 1.G by A3,GROUP_1:def 3;
  then x = a" * 1.G by A2;
  hence thesis by A3;
end;
